Exponents Chapter Two

Exponents - Chapter Two

• Exponents are a mathematical concept representing repeated multiplication
• Rational exponents are exponents in the form m/n, where m and n are integers

Rational Exponents

• Rational exponents represent roots of numbers
• Example: a^m/n = (ⁿ√a)^m
• Special case: ¹/_n√a = a^1/n

Example 1

• Calculating exponents without a calculator
• Example: 27^2/3 = (3³)^2/3 = 3² = 9

Example 2

• Simplifying expressions with rational exponents
• Example: (81x^-3y⁴)/(16x³y⁴) = (81/16) x^-6y⁰ = (27/4) x^-6

Example 3

• Simplifying expressions involving roots
• Example √x³/√x²

Note

• Coefficients in expressions with roots need to have the root applied to them
• Exponents need to be divided by the order of the root
• Example: √16x⁸= 4x⁴

Exercise 1

• List of practice problems

Simplifying Exponential Expressions

• Methods for simplifying expressions using prime numbers and exponent laws

Example 1 (from Ex 2)

• Simplifying expressions with exponential forms.
• Example: 4^*2x^-2/8x^-1 and 5x⁹x+1 /3x⁺²15x^-1

Example 2 (from Ex 2)

• Simplifying and solving expressions involving roots
• Examples: 2√2 √x²/√16x⁴ and √8 x^-2/√9 x⁴ √62x-9/2√6.

Exercise 2

• List of additional practice problems

Addition and Subtraction

• Simplifying expressions involving addition and subtraction of exponents using factorization

Example 3

• Simplifying expressions using factorization
• Examples: 3x⁺¹-3x⁺²/3x^-3-3^-1 and 3^2x+3x/3^x+1

Other Types of Factorization

• Factoring expressions using differences of squares
• Factoring quadratic trinomials
• Examples: 3^2x-9 and 5^2x+5^x+1-6/ 5^x+6

Example 4

• Simplifying expressions; use of alternative methods
• Examples: 4/2⁺¹ and factorizing with rational exponents;

Example 5

• Simplifying expressions with surds;
• Examples: simplifying √48 and simplifying √54

Example 6

• Simplifying expressions with surds;
• Examples: √2+√8 and (√12+√27)² and (√2-√3)²

Rationalizing the Denominator

• Methods for writing fractions with surds in a form WITHOUT SURDS IN THE DENOMINATOR

Example 7

• Rationalise denominators. Example: 5/√2 and 4/(3-√5)

Exercise 5

• List of practice problems for consolidation

Exponential Equations

• Equations in which the variable is an exponent
• Techniques to solving such equations

The Basic Type

• Making all the bases the same, applying exponent laws and equating exponents

Example 7

• Solving for x: Example 3^-2x = 24 and 9^x-1 = 27.3^x+2 and √4=8x¹/√8x

Using Factorisation

• Identifying common factor type equations
• Examples: 2x+2-2^x = 12 and 3(9^-3^2x-1 = 24

The Quadratic Trinomial Type

• Identifying Quadratic trinomial type equations. Example: 9^x-(4)(3^x)+27=0 and 4^x-3(2^x)+4=0

Exercise 3

• List of practice problems

Equations with Rational Exponents

• Solving equations with rational exponents using reciprocal techniques for evaluating

Example 1.

• Solving for x: x^5/3 = 8

Special Cases

• Cases to be aware of when dealing with even powers or negative values inside roots of expressions. Example x²/n = a where m and/or n are even.

Example 2

• Solve for x ( a) x^2,3=9, (b) x^5/2=32, (c) x^1/3 = -2

Example 3

• solving for x; x^1/3 = 3 and x^5/3 = 32

Example 4

• Solving for x; x-3x²+1 = 0 and x²-3x^-1-2 =0

Exercise 4

• List of practice problems